in mathematics, the determination of the set of values of the constants in the solution of a given second-order differential equation that make the solution satisfy not only the differential equation but also a set of specified auxiliary conditions usually called boundary values (see boundary value). The principles of solving this problem were established by the French mathematicians Charles-Franois Sturm and Joseph Liouville in the 1830s; in the 20th century those principles have been applied in the development of quantum mechanics, as in the solution of the Schrdinger equation and its boundary values. A simple example of such a problem is finding a solution y(x) to the equation y + c2y = 0 such that the function equals zero if x is equal to 0 or some number a. The function y = sin cx satisfies the equation, but it meets the auxiliary conditions only if c = np/a, in which n = 0, 1, 2, . . . . These problems are also called eigenvalue problems and involve more generally the problem of finding a solution of the equation + [q(x) - kr(x)]y = f(x) that satisfies the auxiliary conditions a1y(a) + a2y(a) = 0 and a3y(b) + a4y(b) = 0, in which a1, a2, a3, and a4 are constants. To determine when this equation has a solution, the related homogeneous equation is first considered; i.e., the equation with the function f(x) equal to zero. If the functions p, q, r satisfy suitable conditions, then, as in the simpler example above, the equation will have a family of solutions, called eigenfunctions, corresponding to certain values of k, called eigenvalues. Then, if the value of k in the original nonhomogeneous equation is different from these eigenvalues, the problem will have a unique solution. If k equals one of these eigenvalues, the problem will have either no solution or a whole family of solutions, depending on the properties of the function f(x).

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